Let $f : X \to Y$ be a [fill in the blank] morphism of [fill in the blank] complex varieties. Then we have the pushforward $f_! : K(X) \to K(Y)$ which is defined by $f_!(E) = \sum_i (-1)^i [R^i f_\ast E]$, the alternating sum of the higher direct images. Here we take $K(X)$ to mean the $K$-group of coherent sheaves.
On the other hand we can also define $K(X)$ as the $K$-group of $C^\infty$ complex vector bundles on $X$ considered as a real manifold. Then we can define a Gysin map $f_!$ using the Thom isomorphism theorem for $K$-theory.
Which adjectives do I need to fill in the blanks with to make the two notions of $K(X)$ agree?
Which adjectives do I need to fill in the blanks with to make the two notions of $f_!$ agree?
If $X$ is smooth and projective, then any coherent sheaf has a finite resolution by locally free sheaves, so we have a map $K^{alg}(X) \to K^{top}(X)$. On the other hand, I don't think it's true that any $C^\infty$ complex vector bundle has a holomorphic structure, so I don't think there is a map $K^{top}(X) \to K^{alg}(X)$ …
Best Answer
I found a paper which I think answers #2:
Baum, Fulton, MacPherson Riemann-Roch and topological K-theory for singular varieties.
They prove that the algebraic $f_!$ and the topological $f_!$ agree in the case of proper morphisms of (not necessarily smooth) quasi-projective varieties, reducing in several steps to the map from a projective space to a point, see theorem 4.1.
In other words, we have a commutative square
$$\begin{array}{ccc} K^{alg}(X) & \to & K^{alg}(Y) \\ \downarrow&&\downarrow& \\ K^{top}(X) & \to & K^{top}(Y) \end{array}$$